Critical behaviour of the monomer-monomer reaction model with desorption

The study of phase transitions in systems far from equilibrium has aroused the interest of the scientific com-
munity and has been growing progressively in recent years, due to its application in various complex systems,
including the heterogeneous catalysis. A heterogeneous catalysis model well known and used to study phase
transitions is the monomer-monomer reaction model, described by the reaction $A + B \rightarrow AB$. This catalysis
reaction model displays a first order phase transition at the point $y_{B}=1/2$, where $y_{B}$ represents the probability
of a monomer $B$ to arrive the surface. For $y_{B}>1/2$ the system evolves into a saturated lattice by monomers of
the type $B$, and if $y_{B}<1/2$ the system evolves to an $A$ poisoned lattice. Only for $y_{B}=1/2$ the system displays
an active stationary state in which the reaction $A + B \rightarrow AB$ occurs indefinitely. We studied, in the present
work, the critical behaviour of this reaction model on a one-dimensional catalytic surface, and attach to
the monomer $B$ a desorption rate $\alpha$. We used the mean-field approach, at the level of sites and independent
pairs, as well as Monte Carlo simulations, and found a change in the type of transition, from a first order phase
transition to a continuous phase transition. This phase transition occurs between an absorbing state, in which
the lattice is saturated by monomers of the type $A$, and an active steady state, in which the reaction takes place
between the two monomers indefinitely. We found that the critical point $(y_{B_{c}})$ depends on the value of $\alpha$, and
a phase diagram that separates the saturated phase and the active steady state was drawn. The static critical
exponents $\beta$ and $\nu_{\bot}$ was obtained through finite size scaling, confirming the conjecture due to Grassberger. We
also obtained the dynamical critical exponents $z$, by introducing a suitable time variable that is defined as being
the mean time required for the system to become completely poisoned, and $\nu_{||}$, through the time evolution of the order
parameter (fraction of empty sites) at the critical point.